Conclusions

Chapter V: Near Perfect Anti-reflection in Lossless Dielectric Nanocone

5.5 Conclusions

is allowed (magenta line), experimental reflection increases rapidly, much quicker than the transmitted diffraction. This provides reasonable explanation of the similar rapid change of reflection reduction in experiments. This plot also provides clues to the differences in magnitude of the experimental reflection increase after the new diffraction order is allowed. The lowest aspect ratio of 0.65 (green dotted line) sees the greatest reflection increase for k₀ above the free space diffraction, followed by the other samples. Thek0at which the reflection begins to decrease again for each experimental curve in Figure 5.9 matches with the fraction of power that is diffracted to free space. Had it been possible to measure decreasingly small wavelengths for the array of d_bot = 540 nm, it would be expected to display light trapping into higherk₀s since it’s aspect ratio cuts through a region of high transmitted diffraction strength and between peaks in free-space diffraction strength. In fact, taking the ratio of A and B showed that an aspect ratio of 1.08 is roughly ideal for light trapping by maximizing the substrate diffraction strength while minimizing the free space diffraction strength. To target the visible to near-infrared light spectrum between 400 nm and 1000 nm, a cone withd_bot u900 nm, height of 972 nm, andd_topless than 250 nm should be ideal. Last, the periodic nature of the axial reflection modes show up in Figure 5.10B) as this mechanism competes with free space diffraction. Of note is the same periodic maxima along the aspect ratio axis appear with free space diffraction as they did with substrate diffraction, and deserve further investigation.

0 1 2 3 4 5 Aspect Ratio (t/dbot)

0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25

k0 (c/2a)

Substrate Diffraction

0 1 2 3 4 5

Aspect Ratio (t/dbot)

Free Space Diffraction

0%

20%

40%

60%

80%

100%

Figure 5.10: The diffraction strength as percentage of transmitted A) and reflected B) flux in a non-zero mode for the canonical cone with varying aspect ratio. Vertical lines correspond, to the experimental cone array’s aspect ratio, where green dots, orange dashes, and solid blue represent experimental arrays ofd_bot=900 nm, 650 nm, and 540 nm respectively. Horizontal lines are the same from Figure 5.9 as an eye guide for the relevantk₀s of diffraction modes.

due to their availability and relevance to applications such as solar PV. However, the results are presented can be applied to any other lossless dielectric combination.

For example, materials that exactly match the substrate index to the approximate continuous index function should give yet fewer orders of magnitude reflection.

Though realistic perfect combinations of materials exist, non-intuitively, the cone material would have higher index than the substrate; the optimum cone index can be calculated from the approximations discussed. Additionally, because the substrate first order is refractive index dependent, the region of light trapping can be extended by increasing the refractive index of the substrate.

Coupling incident light into the low order diffraction modes that cannot easily trade momentum with the lattice for free space scattering internally traps light between the substrate and absorber, creating multiple absorption opportunities.

This manifests as sharp dip in the apparent reflection reduction of experiments with planar photodiodes. This phenomenon puts low-aspect ratio nanocone arrays into a unique category that provide broadband anti-reflection and light trapping for nearly all AOIs, while also being designed to target specific wavelengths to a even greater degree. One optimal nanocone aspect ratio is found to be 1.08, as it balances strength of substrate diffraction efficiency with minimized free space out-coupling. In experiments with silicon PV cells, surface texture masked effects of diffractive light trapping due to scattering, allowing for parsing of near zero glass surface reflection of less than 0.37% in the visible spectrum. Similarly low

arrays experimentally show minimal to no overall angle dependence. Apparent glass reflection at high AOI was reduced by as much as 21%, the product of angle independent light trapping and waveguide coupling with a smooth effective index function.

Using silica sol-gel and NIL enables promising application of these structures as an affordable and scalable augmentation of existing solar cell designs. Not studied here, but encouraging, other work has shown nanocone surface structures can be designed to have self-cleaning capabilities in addition to being environmentally inert silicon oxide, decreasing solar PV maintenance costs while increasing collected sunlight over time. Due to the redirection of light via diffraction, so called "dead space" in solar modules between cells could potentially be mitigated. Reducing the reflection of the solar cell cover glass would provide an additional 1.42 mA cm⁻²photocurrent and could lead to power conversion efficiency gains of 1% or greater.

C h a p t e r 6

PHOTONIC STRUCTURES TOWARDS PERFECT REFLECTION IN LUMINESCENT SOLAR CONCENTRATORS

6.1 Introduction

Much of current PV research maintains focus on decreasing the levelized cost of solar electricity solar to prices lower than coal, oil, and natural gas. Concentrat- ing photovoltaics (CPV) devices offer a possible solution to minimizing the overall cost of PV-generated electricity. By concentrating the light that impinges upon a solar cell, the device power conversion efficiency can be increased. Addition- ally, by developing a concentration method that focuses the light to a smaller area, the amount of costly PV material required decreases. However, two fundamental hurdles for geometric CPV cells exist. First, the complexity and cost of optics involved in concentration sometimes outweigh the benefit of decreased device area.

Second, CPV cells often require tracking hardware in order to effectively concen- trate light. A LSC overcomes both of these shortcomings of traditional geometric concentrators.[Lunardi2019]

A schematic of the LSC is seen in Figure 6.1. The large light gray rectangle represents a slab of dielectric material, typically a coating low index polymer. Solar cells are demarcated by dark gray rectangles. In the top instance these cells are coupled at the sides to a slab of low index material, edge-lining the slab with 4 solar cells. The second instance (bottom) shows a solar cell that is in the same plane as the waveguide. Both designs have strengths and weaknesses, yet the edge-lined solar cell is more thoroughly studied than the coplanar design. The coplanar design is given careful consideration in the investigations herein. Common to both cases, sunlight is incident from the top of the polymer slab surface. Luminophores, marked as circles, are homogeneously distributed with a controllable concentration. The luminophores absorb and then re-emit light as photoluminescence (PL). Specifically, the luminophores studied here absorb high energy blue light and emit lower energy red light; the benefit of this is treated below. The particle concentration of the dots determines the amount of light absorbed in the slab, yet it also determines how much scattering and potential re-absorption of emitted light the luminophores participate in. The escape cone indicates the angles at which light can enter and leave this

ℓWG

ℓsc(2)

n > 1

Solar Cell

!c

n > 1

… +

+ -

Coplanar

Figure 6.1: A schematic of a single junction LSC shows the solar cell placement, escape cone, luminophores, emission travel path, length L and thickness T, and solar cell width l. Bold red and blue lines indicate electrical connections to the photovoltaics. Figure courtesy of D. Needell

slab of material. Emitted light travels along the paths indicated by the red arrows, showing how rays of luminescence are guided by TIR.

The theoretical benefit of these devices is that the acceptance area can be orders of magnitude greater than the actual photovoltaic device. Using a low index polymer help to accept more diffuse light due to its low reflectance at large AOIs and absorp- tion of luminophores is angle independent. This is contrasted by the large reflection exhibited at large AOIs for typical high refractive index semiconductor PV devices.

Large materials savings could conceivable be attained by swapping a production of large area, high-cost III-V solar cells such as GaAs for a low cost, high volume production of polymers and luminophores like polylauryl-methacrylite (PLMA)and QDs. Only a fraction of the area of the high cost solar device would be needed in such a device. Additional savings come from other auxiliary costs involved in solar energy deployment such as the cost to ship and install heavier crystalline materials versus light weight polymers. Further, the absorption and visual appearance of the device by tuning the luminophore optical properties. Schemes of transparent, en- ergy producing windows have already been demonstrated in experiment.[104] This same transparency allows LSCs are a potential mate for traditional silicon PV cells in a modular tandem design. An LSC that more efficiently converts high energy photons by guiding them to a higher voltage cell allows lower energy photons to pass through to the silicon bottom cell, outputting the combined power.[111] Here, a focus is given to these tandem devices in an exploration of how they might be

Figure 6.2: A mock-up of an InGaP LSC/Si tandem solar cell. Quantum dot luminophores are embedded in a PLMA waveguide. These absorb high energy blue wavelengths and luminesce at lower energy red wavelengths. The longer wavelengths not absorbed by the luminophore pass through to the Si bottom cell.

Top and bottom filters depicted help to trap luminescence and are studied in detail.

Figure courtesy of D. Needell

realized and their ultimate performance limits.

A tandem LSC device is proposed and depicted in Figure 6.2. A PLMA waveguide embeds CdSe/CdS quantum dots and a high performance InGaP micro PV cell.

These micro cells are arranged in a grid along the bottom of the waveguide. The waveguide is sandwiched between two filters that have been optimized to maximize luminescence trapping and power output of the tandem design. The design follows from the methods and analysis presented in the subsequent sections. The goal of these filters is to better trap PL from the QDs that emits within the TIR escape cone.

The top module is separated by an air gap from a silicon bottom cell. The higher bandgap InGaP cells are able to output more power than the lower bandgap Si cell due to less thermalization loss. They thus have a higher power conversion efficiency for blue photons. Red photons below the InGaP band gap are converted by the Si cell. The sum of the two powers exceeds that of either individual cell.

6.2 Analytical Treatment, Investigation, and Insights of LSCs Thermodynamic Concentration Limit

The angle of TIR in Figure 6.1 is given by Snell’s law,θ = sin⁻¹(ⁿ_n²

1). Only light that is outside this escape cone is guided to and collected by cells. Light within the escape cone is typically lost after a few bounces between the top and bottom surface. The Fresnel equations for reflection magnitude of perpendicular(s, TE) and parallel(p, TM) polarizations, are given in 6.1. TE reflection increases monotonically polarized

R_{T E} =

tan(θ_i−θ_t) tan(θ_i+θ_t)

2

,R_{T M} =

−sin(θ_i−θ_t) sin(θ_i+θ_t)

2

(6.1) The mechanism leveraged in an LSC is the absorption and consequent luminescent emission of red-shifted light. This change in energy is known as the Stokes- shift.[152, 182] When a luminophore is embedded in a dielectric slab, luminescent intensity is distributed into a full 4π steradians.[183] The slab acts as a waveguide and concentrator because a significant portion remains trapped in the index guided modes of the slab. Light intensity in the slab is maximized by accessing all available modes of the slab. Relative to free-space, this intensity is greater by a factor of 2n² within the slab.[183] This process is similar to the enhancement of Figure 1.4.

Coupling a solar cell to this slab gives the trapped photons an escape route. In a lossless system every photon eventually finds the solar cell and is absorbed. In this fashion, the slab’s area relative to the solar cell can be many factors greater. The same photon flux as is incident on to the slab will reach the solar cell as it travels the waveguide. The ratio two areas and is known as the geometric gain (GG). This is represented in 6.1 by the ratio of LandT in the first case and the ratio ofLandl in the second case.

The theoretical power conversion efficiency,η, limit of PV cells coupled to an LSC has been shown to not be greater than, but equal to, to that of a non-concentrating PV device with an area equivalent to waveguide.[56, 124] [102] Traditional CPV concentrates by decreasing entropic loss through restricting the etendue of a solar cell; is no photon energy loss. In LSCs the down conversion of the absorbed high energy photons is analogous to a trade of energy for for the reduced entropy, that is, concentrated light intensity. There exists a fundamental thermodynamic relation between the achievable concentration in a waveguide and the magnitude of the Stokes-shift.[182] The limit of the concentration is defined as the ratio of the external brightness,B_ext to internal brightnessB_int,C = ^B_B^ext_int, where Bhas units of intensity per unit area per solid angle per second. For an LSC,C is approximated by Equation 6.2 whereω₁andω₂ are the frequencies of absorption and emission, respectively.

C ≤ ω²₂ ω²₁e

~(ω1−ω2)

kbT (6.2)

Equation 6.2 can be re-written in terms of energy as approximately[145]:

C ≤ E₂³ E₁³e

(E1−E2)

kbT (6.3)

Equation 6.3 can be used to calculate the maximumC achievable by a terrestrial based LSC with a luminophore that perfectly absorbs up to a certain energy E₁ and perfectly emits at a lower energy E₂. This takes into account the infinitesimal fractional photon flux at E₁relative to the total AM1.5G solar spectrum flux I(E) asδf_E1. Hereδf_E1is the Kronecker delta function used to select only the fractional portion of the spectrum atE₁.

C ≤ Z ∞

0

δE,E₁f_E1E₁³ E₂³e

(E1−E2)

kbT dE (6.4)

This integration accounts for the fact that there are less high energy photons in the solar spectrum. The Stoke’s shift represents a band of photons that are lost. So, there must be a balance between flux concentration to the flux that is lost due to the energy gap.

Concentration values in Figure 6.3 are affected by the solar spectrum photon dis- tribution to a degree but the exponential trend is similar. The relationship for an evenly distributed spectrum would be symmetric with energy. These values also do not necessarily imply that an LSC should exceed the efficiency of a single junction since that is thermodynamically determined by detailed balance. The optical envi- ronment for cells coupled to a waveguide inherently changes their external radiation efficiency, i.e. their radiative recombination rate.[143] Figure 6.3 highlights the importance of choosing the correct, and large Stokes-shift for a luminophore in an LSC. The largestCfactors demonstrated to date are only on the order of 30x relative to specific bands of the solar spectrum.[13] Concentration factors were only on the order of 20x prior to recent use of large Stokes-shift luminophores. The power conversion efficiency of full LSC PV devices has been correspondingly small also.

Figure 6.3: The concentration limit of luminescent solar devices given by Equation 6.4 relative to the entire solar spectrum. The concentration factor greatly exceeds the geometric concentration limit of 42,600 suns.[119]

Analytical Model of Photon Travel Path in a TIR Waveguide

The large disparity between theCfactors achieved thus far and the correspondingly low efficiencies arises from a variety of loss mechanisms. Equation 6.5 give the expression for the maximum power output of a solar photovoltaic cell. The V_OC, J_SC, and FF are the device voltage at open circuit, the short circuit current density, and the fill fraction at the maximum power point, respectively. The power conversion efficiency of the overall device is given as the fraction of incident power to the output power in Equation 6.6.

P_max =V_OCJ_SCF F (6.5)

ηdevice = Pout

P_in = VOCJSCF F

P_in (6.6)

This work focuses on increasing the J_SCcomponent first, though many strategies to affect the J_SCcan also positively affect the other two factors. The J_SC for a is found by the integral of the entire frequency dependent

J_SC =Z ∞ ω=0

I(ω)η_coll(ω)dω (6.7)

where I(ω) is the intensity and η_coll(ω) is the collection efficiency of the photo- converting device, in this case a photovoltaic cell. The collection efficiency in a typical solar cell is governed by its own properties alone. The generalized function η_coll(ω) includes consideration of other less-traditional mechanisms, such as the incorporation of targeted band rejection filters and up or down frequency shifting elements such as luminophores with a large Stokes-shift. In an LSC, luminophore α, PL, and the waveguide loss must be taken into account as in Eq. 6.8.

η_coll(ω, µ,T) = α(ω, µ,T)η_PL(ω)η_Guide(ω)η_EQ(ω) (6.8) In Eq. 6.8,α(ω, µ,T),η_PL(ω),η_Guide(ω),η_EQ(ω), correspond to the luminophore absorption probability, the photoluminescent quantum yield (PLQY), the waveguid- ing efficiency, and the photovoltaic external quantum efficiency respectively. The independent variablesω,µ, andTare the frequency, chemical potential, and temper- ature. Typically, the chemical potential and temperature dependence of absorption of the the quantum dots is a second order effect and it is a reasonable assumption that α(ω, µ,T) → α(ω). However, if sufficiently large concentrations of light are achieved in an LSC, these factors, especially µ, become increasingly important.

Each of these factors will depend on the optical design of an LSC to different degrees.

A novel analytical expression for the average distance a photon travels in a waveguide was developed in order to describe the loss from a slab waveguide. This is used to determine the intensity I(ω) seen by a collector in the waveguide. The aim is to fully account for the parameters and properties of real devices. Important factors include placement of solar cells, losses such a non-unity PLQY, the EQE of the solar cells. Design choices, such as the use of notch filters should also be accounted for. Optimization of an increasing number of parameters in the complex LSC system is difficult. Modeling of LSC loss mechanisms is typically done via Monte-Carlo ray tracing. This has proven to be accurate on many occasions, but is computationally expensive.[124] [12] [11] [179][123] Simplified descriptions using fundamental equations that describe the behavior of LSCs allows greater optimization opportunity. To date few others have attempted to supply such a model, and a great need is unfilled.[117] The complexity of the device encourages a modular approach when developing these types of mathematical descriptions.

The different loss mechanisms of the LSC in Figure 6.5 were considered. The closed

Figure 6.4: The typical EQE, Reflection, QD luminophore absorption and PLQY spectra for the components of a tandem LSC studied. The AM1.5D photocurrent spectrum is plotted in the background. The QD absorb blue light and emit PLnear 630 nm. This is matched to the InGaP EQE. The Si bottom cell EQE (purple) is most efficient in the infrared portion of the spectrum. A dielectric notch filter reflection spectrum is shown to have been designed to perfectly reflect light in the PLspectrum and have now reflection outside of this band (black). Figure Courtesy of D. Needell

form analytical model accounts for each mechanism and calculates a J_SCas an input to the detailed balance model of Shockley and Queisser to find the corresponding V_OC and FF of 6.5. [143] The intensity distribution of PL within the waveguide around the collector is defined asI_PL. This intensity is determined by 6.7.

Equation 6.9 defines a parameter N as the average number of reflections from the top and bottom guiding surfaces as a function of angleθ from the normal vector of the waveguide, the average scattering distanceξ, and the thickness of the waveguide T. The average scattering distance ξ is related to the scattering of waveguide imperfections, such as particulates, and the overlap of the absorption and emission probabilities of the luminophore. ThicknessT is assumed to be constant through the rest of the derivation, but note here that it can be varied. Equation 6.9 derives from 2-dimensional trigonometry. It assumes each photon begins at the bottom of the waveguide, slightly overestimating the actual distance traveled. The upper limit on this error is a single integer multiple of N.

N(θ, ξ,T) = ξ

T cos(θ) (6.9)